\chapter{Plan of Remaining Research Tasks}
In this chapter I provide a summary of the work that has been done so far and discuss my plan for remaining research tasks. The proposed plan of work consists of four parts and are supposed to complete the thesis.

\label{chap-five}
\section {Summary of the work already conducted}
In this proposal we undertook a study of capacity design and expansion in failure-prone networks. We assumed a network faces many non-simultaneous failure scenarios, affecting capacities and demands. The design aimed to protect the demands against all possible failure scenarios, e.g., survivable network design. Our specific contributions are as follows:
\begin{itemize}
  \item We introduced an integer programming model for the capacity design in survivable networks
  \item Our approach to modeling and incorporating failure scenarios into our mathematical formulations is unique.
  \item We developed methods that lead to the reduction of the solution space. Theses methods guarantee to keep the global optimal solution in the search space. Therefore the reduced integer programming model is guaranteed to find the optimal solution;
  \item We provided methods for finding upper bounds and lower bounds for our cost function.
  \item We developed a decomposition algorithm that solves a sequence of stronger integer programming models, rather than solving the problem directly. The algorithm guarantees to find the optimal solution.
  \item We conducted experimental evaluations of our algorithm on problem instances with specific characteristics and presented the numerical results.
\end{itemize}
Our experiments show that the proposed algorithm is effective at least in many realistic size problem instances with a single failure scenario. We expect the algorithm to be more effective as the size of test instances grow, specially in the number of possible failure scenarios. More experimental evaluations are needed to study this hypothesis.

\section {Plan of Future Work to Complete the Thesis}
In the immediate future I plan to pursue the following research activities to complete my thesis.
\subsection {Small Algorithmic Improvements}
The implemented algorithm can be improved in some ways, including warm-starting the master problem each time it is called. I am planning to implement this idea immediately and study its effect on the efficiency of the algorithm.
\subsection {Experimental Evaluation on Models with Multiple Failure Scenarios}
As discussed in the previous chapter, the experiments that have already been conducted only include the models with single failure scenario. As the proposed algorithm decomposes the problem by failure scenario, we expect more improvements in solution time when the number of failure scenarios increase. However this hypothesis should be tested.
\\
\indent{}Moreover, the experiments already conducted are all with fixed demands. Experimental evaluations need to be designed and performed with varied levels of point-to-    point demands.
\subsection {Implementing Methods to Reduce Solution Space and a Warm-Start Option}
In chapter \ref{chap-two} we introduced methods to reduce the number of variables and constraints of our problem. These reductions are expected to contribute to the improvement of solution approach, while guaranteeing that the global optimal solution is not being cut off. As a part of the future work, I plan to implement these methods and evaluate their effect on the efficiency of solution approach using test instances.
\\
\indent{}One specific method that I plan to implement is  providing a good warm start solution. By finding and using a good warm start solution (rather than starting to solve the master problem at the trivial solution of zero), we expect to have a more time-efficient solution approach. Please note that this warm start is different than the one already discussed in the "Small Algorithmic Improvements" section that solves the master problem each time. The new approach will use a good starting point for the whole problem.
\subsection {Implementing a multi-cut approach}
In the proposed algorithm, each time a cut is generated, it is passed to the master problem. Master problem (which is an integer model) is solved once and its optimal values get updated and passed again to the sub-problems. This means that for each cut that is generated, master problem needs to be solved once. This approach might be applicable to the problems where the size of the master problem is rather small; for example a network with a rather small number of links and capacity expansion alternatives. However, if the network is larger, solving the master problem several times would be costly. For this reason, an approach that reduces the number of calls to the master problem might be helpful. One possible way to implement such an approach is to solve the sub-problems associated with a set of failure scenarios at the same time using the same optimal solution ($\bar{z} $) provided by the master, and then passing an aggregated cut, or a number of separate cuts to the master problem at the same time. As this approach does not need to update $\bar{z} $ after every single cut, it reduces the number of calls to the master problem. This of course comes at the price of solving larger master problems each time. Therefore this idea needs to be investigated more.

Table \ref{tab:one} provides a time line for my plan to finish the mentioned research activities and complete the thesis. The tentative time to finish up all activities is 36 weeks.

\begin{table}

\caption{Schedule of Future Research Activities to Complete the Thesis}

\label{tab:one}

\begin{center}

\begin{tabular}{|l|l|c|}

\toprule

Activity No. & Description  & Time (week) \\
\hline\hline
5.2.1 & \textbf{Small Algorithmic Improvements}&\textbf{5}\\
\hline
& Designing and implementing the algorithmic changes&2\\
\hline
& Evaluating the effect of changes on the algorithm&1\\
\hline
&Organizing the findings and preparing a report&2\\
\hline\hline
5.2.2&\textbf{Experiments on Models with Multiple Failure Scenarios}&\textbf{7}\\
\hline
&Completing the code to handle models with multiple failures&2\\
\hline
&Performing experiments to evaluate the extended algorithm&3\\
\hline
&Organizing the findings and preparing a report&2\\
\hline\hline
5.2.3 & \textbf{Implementing Reduction Methods and a Warm Start Option}&\textbf{8}\\
\hline
&Implementing solution space reduction methods&2\\
\hline
&Designing and implementing the warm start&2\\
\hline
&Evaluating the effect of changes on the performance of the algorithm&2\\
\hline
&Organizing the findings and preparing a report&2\\
\hline\hline
5.2.4&\textbf{Implementing a Multi-Cut Approach}&\textbf{10}\\
\hline
&Designing a multi cut algorithm based on the problem structure&2\\
\hline
&Implementing the designed algorithm&3\\
\hline
&Conducting experiments to evaluate the performance of the algorithm&3\\
\hline
&Organizing the findings and preparing a report&2\\
\hline\hline
&\textbf{Preparing for Final Exam}&\textbf{6}\\
\hline\hline

\end{tabular}

\end{center}

\end{table}

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